An integral aspect of finite element methods is to find the solution of the corresponding linear system of equations. For small systems this is generally done by a direct solver, which factorizes the linear system and directly computes the solution up to machine precision. The computational cost of direct solvers scales poorly with the size of the system, however, resulting in computation time and memory requirements that become prohibitive for large systems [1]}. For this reason, large systems are generally solved by iterative solvers, the computational cost of which generally scales better with the size of the system [2]}. The convergence of iterative solvers is strongly correlated with the conditioning of the system. Without tailored stabilization or preconditioning, systems derived from immersed finite element formulations are generally severely ill-conditioned, impeding the application of iterative solvers [3]}. In Section REF , an analysis of the causes of ill-conditioning in immersed finite element systems is presented. Preconditioners which effectively resolve these conditioning problems are discussed in Section REF . Section REF  provides a detailed discussion about Schwarz preconditioners, which have commonly been applied to immersed finite element systems in recent years. This last subsection also includes an example that illustrates the effect of Schwarz preconditioning on an immersed finite element simulation of a scan-based geometry.
